An arbitrary Lagrangian-Eulerian finite element method for finite strain plasticity

This paper presents a new arbitrary Lagrangian–Eulerian (ALE) nite element formulation for nite strain plasticity in non-linear solid mechanics. We consider the models of nite strain plasticity de- ned by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part (F=FeFp), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very e ciently with the use of the connectivity graph of the xed reference mesh. A staggered scheme de ned by three steps (the smoothing, the advection and the Lagrangian steps) leads to an e cient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi-static and dynamic conditions are considered in these model examples